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Dear Professor Mean, I need to randomize the order in which I give treatments and controls in my research study, but I don’t know how to randomize. Can you show me what to do? — Baffled Beth

Dear Baffled,

Here’s a lightbulb randomization joke: take to in How many a screw does statisticians it lightbulb?

There are also two quotes that are relevant here: “Random selection is too important to be left to chance” and “Random is not haphazard“. The latter quote can be found in Statistically Speaking. A Dictionary of Quotations by C.G. Gaither and A.E. Cavazos-Gaither.

Short answer

The simplest way to randomize is to

list your treatments and controls in a systematic order,

attach a random number to each row, and

sort by the random number.

This works for simple randomization, block randomization, and randomization in a cross-over design.

Why randomization is important.

I’m glad that you are interested in randomization. Some people think it is fine just to alternate regularly between treatment and control. But there are two problems with this.

First, if you’ve watched plants grow in a garden, you will notice that they tend to have a large-small-large-small pattern, especially if they are planted too tightly together. The plants are competing for resources. If one plant gets a small lead, its increased consumption will make the plants on either side of it grow more slowly. If you alternated between treatment and control in this garden, you would notice a difference that is entirely independent of the treatment.

This same sort of large-small pattern can occur in medical situations. For example, if a doctor spends too much time with his/her first patient, it increases the probability that the next patient will get less time (in order to keep things on schedule).

A second problem with alternating is that many studies are blinded. An alternating schedule may make it easy for someone to guess what patients get what treatment, ruining the blinding.

Example

Let me show you this in Excel, but the general method works well for any program that can generate random numbers. Suppose we have a study with twelve patients, six treatment and six control. List C and T alternating down in the first twelve rows of the first column of the spreadsheet. In the second column, place the RAND() function in each of the first twelve rows. Here is what you have:

Your random numbers will be different than these. Otherwise, they wouldn’t be random now, would they?

To randomize, sort the data by the second column. Notice that the systematic order of the first column has been scrambled.

By the way, when you try this on your spreadsheet, you will notice that after the sort, the random numbers are recalculated. This seems a bit disconcerting, since the sorted numbers randomly change to different (and unsorted) values. You had to look really, really fast to see the sorted numbers.

When I work out these examples, I freeze the random numbers by pasting their values only into a third column (select EDIT | PASTE SPECIAL) and then deleting the second column. That’s why the random numbers in my example stayed sorted. It doesn’t make much difference, though, what happens with the second column; it’s the first column we are interested in.

What is block randomization?

If there is the possibility that the experiment will not reach your target sample size, then sometimes randomization can leave an imbalance due to the (bad) luck of the draw. For example, if the previous experiment ended halfway, we would have had four treatment patients and only two controls. This is not too terrible, but for a larger sample, the discrepancies can get quite problematic.

A solution to this problem is to divide your study up into blocks and then randomize within each block. If the study ends at a block boundary, then you are guaranteed to have perfect balance between treatments and controls. If a study ends in the middle of a block, there might be some imbalance, but probably much less than with regular randomization.

In Excel, we would arrange the T and C values systematically and then add a column with a block number. The third column would be our random numbers. Here is what it would look like.

Now we sort the numbers, by block and by random number within block.

Notice that the first block has the order TCTC, the second has CTCT, and the third has CCTT.

How do I randomize in a matched pairs study?

You can randomize a matched pairs study the same way as with block randomization. In a matched study, every patient gets both the treatment and the control (separated by enough time so that the effect of one cannot carry over into the other). It is important to randomize the order in which the patients get the treatment and the control.

The easy way to do this is to allocate two rows for each patient and label the rows T and C in a systematic order. Add random numbers and then sort within each patient.

Here’s an a example of how we would set up a matched pairs example for six patients.

Sort the data by patient and by random number within each patient.

Notice that patients 1,3, and 4 get the control first and patients 2, 5, and 6 get the treatment first.

Assigning random id numbers for anonymization

Suppose you want to create an identification code that is effectively random. You want to avoid medical record numbers because they would compromise security. There are several approaches that work. A simple approach is to list the numbers 1001, 1002, etc., but this gives a hint as to when the subject joined the study. A better approach is to list those numbers and then sort them in a random order. These numbers can then be cut-and-pasted next to the medical record numbers.

Here’s the medical record number, a sequential ID, and a column of random numbers.

Sort the last two columns only to get a shuffled list of IDs.

Keep this list in a secure location. Use only the ID column when sending your data out beyond the hospital. This approach protects the privacy of individual subjects.

Whenever you are concerned about privacy in a research study, it is a good idea to discuss your concerns with your IRB (Institutional Review Board).

Update

In an August 10, 2005 weblog entry, I discuss how the use of randbetween() function leads to problems with a random amount of imbalance being created. For example, the spreadsheet I was given had 223 people randomized to the first group and only 177 randomized to the second group. This is well within sampling error, but still leads to a slight inefficiency in the analysis.

Summary

Baffled Beth does not know how to properly randomize subjects for her research study. Professor Mean explains how randomization differs from allocating the treatment and control in an alternative fashion. He shows a simple approach for randomization, which involves

arranging your treatments in a systematic order,

attaching a column of random number to this list, and

sorting by the random number column.

He then shows examples for simple randomization, block randomization, and randomization in a matched pairs study.

Further Reading

There’s an interesting two page article in the British Medical Journal on randomisation (notice the alternate spelling. I feel so intellectual when I replace all my z’s with s’s.)